By Dahlquist G.
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Additional resources for 33 years of numerical instability
In other words, note that Eq. (16) dictates that, except for the possible addition of a trivial constant, Po determines the one-body multiplicative attractive potential, v. Also observe that Eq. (12) requires that  (17) for arbitrary g. It is anticipated that the above inequality, the convexity constraint, shall prove useful for generating approximations to F in the future. 7 3. ) sVt i=1 It then follows from the Hohenberg-Kohn proof for the above corresponding interacting situation that E.
Moreover, the fact that the external potential does not appear explicitly in F[p] does not mean that it does not affect the wavefunction. The reason is that the external potential determines the density and the latter is in a one to one correspondence with the wavefunction. e, general expressions containing only p) but also other modulating factors that are specific to a given system. These modulating factors, of course, become constants when the system considered is a homogeneous electron gas.
On the other hand, the oompli~ated scaling for E c arises from the fact that the constrained-search definition of 'I' in Eq. (35) involves two operators with different homogeneous proper~ies; T is homogeneous of degree -2, because it contains second derivatives, while Vee is homogeneous of degree -1 as stated above. ) Equation (39) is equivalent, at A. = 1, to ;nln Ec[p] = (42) £c,i[p] ~ i=2 where £c,i is homogeneous of degree i-2. i -2 £c,i[P]. (43) Moreover, £c 2[P] is known exactly as the second-order contribution  in a recently developed DFT perturbation theory [14,25].